Vector control of an induction motor

ABSTRACT

The invention relates to a method in connection with an inverter, the inverter comprising a direct voltage intermediate circuit, an optimum switching table and output power switches. The method comprises steps where phase currents (i A , i C ) of the inverter are converted to a synchronous dq coordinate system in order to achieve vector components (i d , i q ), the synchronous current vector components (i d , i q ) are low pass filtered in order to achieve current vector components (i d,jpf , i q,jpf ), a current reference (i q,ref ) is generated in the direction of the q axis, a current reference (i d,ref ) is generated, a torque reference (t e,ref ) is generated, an absolute value reference (|ψ| ref ) of a flux linkage is generated from the currents and switching commands are formed on the basis of the torque reference (t e,ref ) and the flux reference (|ψ| ref ) using the optimum switching table ( 5 ).

BACKGROUND OF THE INVENTION

The invention relates to a method in connection with an inverter, theinverter comprising a direct voltage intermediate circuit, an optimumswitching table and output power switches.

A frequency converter is a device that is typically used for motorcontrol. A frequency converter typically consists of two converters,between which there is a direct voltage or direct current intermediatecircuit. The converters of a frequency converter can be implemented insuch a way that they are capable of functioning only as rectifiers, orin such a way that they can function, if required, as both rectifiersand inverters. An example of rectifiers is a diode bridge, and anexample of a converter applicable to both rectification and inversion isa converter bridge implemented by means of transistors. An inverter istypically used to control the power transferred from the intermediatecircuit of a frequency converter to a motor. By means of an inverter,motor control can be implemented reliably in such a way that the motorimplements for example the desired speed or torque command accurately.An inverter can also be used for controlling the power flow from theelectrical power network to the intermediate circuit of the frequencyconverter. An inverter used for this purpose is usually called a networkinverter. A network inverter allows efficient control of the active andreactive powers transferred between the electrical power network and thefrequency converter.

A network inverter is used for replacing a diode bridge rectifier of afrequency converter, particularly in such objects of use where it isdesirable to invert the braking energy of a motor back to the electricalpower network. The curve form of the supply current of the networkinverter can be made very sinusoidal, owing to which it is wellapplicable to objects where the lowering of the electricity qualitycaused by the frequency converter must be reduced.

Present high-rate power semiconductor components and signal processorsmakes it possible to implement the control of an inverter dynamicallywith a high rate by using solutions based on direct torque control(DTC). A known network inverter based on DTC control is shown in theblock diagram of FIG. 1. In the solution of FIG. 1, the switchingcommands of a semiconductor power switch bridge is formed in accordancewith the DTC principle on the basis of the absolute value of the fluxlinkage vector and of the torque by using a DTC block 13. Thecomputational flux linkage vector of the network inverter is calculatedwith a voltage integral{overscore (ψ)}=∫ūdt,  (1)

and the torque proportional to the power is calculated by the crossproduct of the power vector and the flux linkage vectort _(e)=|{overscore (ψ)}×ī|.  (2)

An intermediate circuit voltage controller 11 generates the torquereference t_(e,ref) on the basis of the difference between the measuredintermediate circuit voltage and intermediate circuit voltage reference.The absolute value reference |{overscore (ψ)}|_(ref) of the flux linkageis generated by means of a reactive power controller 12 by comparing theestimated reactive power q_(est) and reactive power reference q_(ref).In connection with network inverters, a low pass filter 14 is typicallyused between the inverter and the network. When the filter type is an Lfilter, the reactive power is estimated with the equationq _(est)=({overscore (ψ)}_(v) ·ī)ω,  (3)where ω is the electric angular frequency corresponding to the directwave of the network, and {overscore (ψ)}_(v) is the flux linkage vectorof the network. The flux linkage vector of the network is estimated withthe equation{overscore (ψ)}_(v)={overscore (ψ)}−L ī,  (4)where L is the inductance of the network filter.

Conventionally, the object of application of power vector controlmethods has been control of electric motor use provided with separatePWM modulators. The principle of power vector control of electric motoruse provided with a PWM modulator is shown in FIG. 3. Power controllers31, 32 generate a voltage vector reference, the intention being toimplement the reference by means of a PMW modulator 33 controlling powerswitches 34. The solution is disclosed for instance in the publicationHarnefors, L., ‘On Analysis, Control and Estimation of Variable-speeddrives’, Doctoral dissertation, Part I, page 44, Royal Institute ofTechnology, Stockholm, Sweden, 1997.

A direct DTC control method of a network inverter similar to that inFIG. 1 does not actively control the currents of the converter. As aresult, the currents of the converter may be non-sinusoidal and containsignificantly lower harmonic components, such as 5^(th) or 7^(th)harmonic. The harmonic current components supplied by the converter areparticularly intensified if, instead of an L filter, the network filteris an LCL filter.

BRIEF DESCRIPTION OF THE INVENTION

An object of the invention is to provide a method by means of which theabove-mentioned disadvantages are avoided and to enable control to anetwork inverter more accurately than before by using the samemeasurements as previously. The object is achieved with a methodaccording to the invention, characterized by the method comprising thesteps of

-   -   defining a synchronous rectangular dq coordinate system;    -   measuring the phase currents of the inverter outputs;    -   converting the phase currents of the inverter to a synchronous        dq coordinate system in order to achieve synchronous rectangular        current vector components;    -   low-pass-filtering the synchronous current vector components in        order to achieve low-pass-filtered current vector components;    -   generating a current reference in the direction of the q axis of        the dq coordinate system;    -   generating a current reference in the direction of the d axis of        the dq coordinate system;    -   generating a torque reference by comparing the current reference        in the direction q with a low-pass-filtered current vector        component in the direction q;    -   generating an absolute value reference of the flux linkage by        comparing the current reference in the direction d with a        low-pass-filtered current vector component in the direction d;        and    -   forming switching commands of the power switches on the basis of        the torque reference and the flux reference by using an optimum        switching table.

The method according to the invention is based on the idea that theoutput current of the network inverter is controlled actively, as aresult of which the control method according to the invention provides anetwork current that is significantly more sinusoidal than before. Inthe method according to the invention, the current controllers generatean absolute value reference of the flux linkage and a torque instead ofa voltage vector reference. Separate voltage vectors are furtherselected on the basis of said reference variables with a DTC algorithmby using an optimum switching table.

The control method according to the invention uses no additionalmeasurements, nor does it set any special requirements compared with thedirect DTC control method. Deviating from conventional current vectorcontrol methods using a separate PWM modulator, the method according tothe invention uses a DTC algorithm in the selection of the voltagevector.

BRIEF DESCRIPTION OF THE FIGURES

The invention will now be described in more detail by way of preferredembodiments, with reference to the attached drawings, of which:

FIG. 1 shows a block diagram of direct DTC control to a network inverter(prior art);

FIG. 2 shows a block diagram implementing a method according to theinvention; and

FIG. 3 shows a principled diagram of current vector control of anelectric motor controlled by a PWM modulator (prior art).

DETAILED DESCRIPTION OF THE INVENTION

A block diagram implementing the method according to the invention isshown in FIG. 2. The control structure is of cascade control type, thecontrol consisting of control loops within each other. The outermostcontrollers are an intermediate voltage controller 1 and a reactivepower controller 2. The intermediate voltage controller generates acurrent reference i_(q,ref) in the direction q, and the reactive powercontroller 2 generates a current reference i_(d,ref) in the direction d.The inner controllers are current controllers 3 and 4 functioning in asynchronous rectangular dq coordinate system and generating an absolutevalue reference |{overscore (ψ)}|_(ref) of the flux linkage and a torquereference t_(e,ref) for the selection algorithm of a DTC voltage vector.The innermost control loop, which is included in the DTC block 5 in FIG.2, is formed of hysteresis control of the absolute value of the fluxlinkage and of the torque.

In accordance with the space vector theory, the variables of athree-step system can be indicated by one rotating vector (e.g. vector{overscore (s)}), which can be divided into rectangular components(S_(d), S_(q)). The above-mentioned synchronus dq coordinate system thusrefers to a coordinate system which rotates along with an electronicproperty and being bound thereto. Typically, a synchronous coordinatesystem is bound to a rotating flux linkage vector, for example. In sucha case, the direction of the d axis of the coordinate system isdetermined to be the direction of said flux linkage vector, and thedirection of the q axis is determined to be 90 degrees from thisdirection.

In accordance with the method of the invention, a synchronousrectangular dq coordinate system is determined. The synchronouscoordinate system is preferably selected in such a way that the d axisof the coordinate system is parallel to the flux linkage vector{overscore (ψ)} of the network inverter. The direction of the fluxlinkage vector is determined by using for example a precise model of amotor, which can also be applied in connection with a network inverterwithout a physical motor. Such a motor model is typically included inthe DTC algorithm so that the direction of the flux linkage vector isprovided simply by using the model. Another alternative for determiningthe flux linkage vector is to calculate it by using the voltage integralof model (1). A synchronous coordinate system rotating at the angularspeed of the network can be bound to other variables instead of the fluxlinkage vector of a network inverter, such as to the flux linkage vectorof the network or the voltage vector of the network. In such a case,however, additional measurements may have to be used.

In accordance with the method, the phase currents of the outputs of theinverter are measured. Measurement of two phase currents are sufficientto determine the current vector if it is assumed that there is noneutral conductor in the alternate current system. Thus, FIG. 2 showingthe method according to the invention only indicates measurement ofcurrents i_(A) and i_(C) of phases A and C.

In accordance with the invention, the measured phase currents i_(A) andi_(C) are converted to a synchronous dq coordinate system. Thecoordinate system conversion is preferably performed in such a way thatthe measured phase variables are converted with a coordinate systemconversion member 6 at first to a stationary xy coordinate system, inwhich the current vector can be denoted by rectangular components i_(x)and i_(y). The measured currents i_(x) and i_(y) are further convertedwith a coordinate system conversion member 7 at the angular frequency ofthe network to a rotating synchronous dq coordinate system. Thecoordinate system conversion from the xy coordinate system to the dqcoordinate system can be performed by means of trigonometric functions

 i _(d) =i _(x) cos θ+i _(y) sin θi _(q) =−i _(x) sin θ+i _(y) cos θ,  (5)where θ is the angle between the stationary coordinate system and thesynchronous coordinate system. In coordinate system changes, the voltageintegral, i.e. the flux linkage {overscore (ψ)}, can also be utilized byusing for the conversion the formulae $\begin{matrix}\begin{matrix}{i_{d} = \frac{\overset{\_}{\psi} \cdot \quad\overset{\_}{i}}{\overset{\_}{\psi}}} \\{i_{q} = {\frac{\overset{\_}{\psi}\quad \times \quad\overset{\_}{i}}{\overset{\_}{\psi}}.}}\end{matrix} & (6)\end{matrix}$where ī=i_(x)+i_(y). In this case, calculating trigonometric functionsand finding out the precise angle of the flux linkage of the networkinverter are avoided. In the formula (6), the cross product is to beunderstood as scalar operator {overscore (ψ)}×ī=ψ_(x)i_(y)−ψ_(y)i_(x),because the direction of the vector generated by the cross product hasin this case no meaningful physical interpretation.

Changes in the coordinate system can be made in a plurality of ways, twoof which have been shown above in equations (5) and (6). In connectionwith coordinate system changes, also other signal processing operations,such as low-pass filtering, can be performed.

The torque of the network inverter can be calculated as the crossproduct of the flux linkage vector and current vector. Indicated in thesynchronous coordinate system, the expression of the torque ist_(e)={overscore (ψ)}×ī=ψ_(d)i_(q)−ψ_(q)i_(d).  (7)

In the dq coordinate system bound to the flux linkage of the networkinverter, the flux linkage has naturally no component in the directionof the q axis. Thus, when operating in the dq coordinate system, ψ_(q)=0and ψ_(d)=|ψ|, whereby the expression of the torque can be indicated ast_(e)=ψ_(d)i_(q)=|{overscore (ψ)}|i_(q),  (8)so that the torque of the converter can be adjusted with a current inthe direction q.

The reactive power of the steady state of the network inverter in thecase of an L filter can be calculated with the formulaq=({overscore (ψ)}_(v) ·ī)ω,  (9)where the flux linkage vector is{overscore (ψ)}_(v) 32 {overscore (ψ)}−Lī.  (10)

Positioning the equation (10) in the equation (9) gives $\begin{matrix}\begin{matrix}{q = {\left( {\left( {\overset{\_}{\psi} - {L\overset{\_}{i}}} \right) \cdot \quad\overset{\_}{i}} \right)\omega}} \\{= {\left( {\left( {\overset{\_}{\psi}\quad \cdot \quad\overset{\_}{i}} \right) - {L\left( {\overset{\_}{i}\quad \cdot \quad\overset{\_}{i}} \right)}} \right)\omega}} \\{{= {\left( {\left( {\overset{\_}{\psi}\quad \cdot \quad\overset{\_}{i}} \right) - {L{i}^{2}}} \right)\omega}},}\end{matrix} & (11)\end{matrix}$which, as shown in the synchronous coordinate system, isq=(ψ_(d) i _(d)+ψ_(q) i _(q) −L|ī| ²)ω.  (12)

When operating in the synchronous coordinate system bound to the fluxvector of the network inverter, ψ_(q)=0 and ψ_(d)=|{overscore (ψ)}|,which givesq=(|ψ|i _(d) −L|ī| ²)ω.  (13)

Since the inductance L of the network filter is usually small, themeaning of the latter term of the equation (13) is small compared withthe first term. Thus, the effect of the current in the direction q onthe reactive power is small, and the reactive power of the networkinverter can be adjusted with a current in the direction d.

In accordance with the invention, a current reference i_(q,ref) in thedirection of the q axis and a current reference i_(d,ref) in thedirection of the d axis of the dq coordinate system are furthergenerated. The current reference i_(d,ref) in the direction of the daxis is preferably determined by using a reactive power controller 2.The task of the reactive power controller is to generate a reference ofa current in the direction d that remains almost constant, with whichreference the desired power factor is achieved for the network inverteras described above. There are no significant dynamic performancerequirements for the reactive power controller 2. Since the currentreference in the direction d that keeps the power factor of the networkinverter one is very close to zero, it is possible to leave out thereactive power controller altogether and to use a fixed reference valueof the current in the direction d instead, or to calculate a suitablereference value with an open circuit equation.

In accordance with a preferred embodiment of the invention, a currentreference i_(d,ref) is generated by comparing the estimated reactivepower q_(est) with the reactive power reference q_(ref). The comparisonis performed in such a way that an estimated value is subtracted fromthe reference. The difference variable provided as the result of thesubtraction functions as the input of the reactive power controller, anda current reference i_(d,ref) is obtained from the output of thecontroller.

The intermediate circuit voltage controller 1 generates a referencevalue i_(q,ref) of the current in the direction q. The intermediatecircuit voltage controller 1 is the most important controller of thenetwork inverter, and together with a current controller 3 in thedirection q, it determines the performance of the intermediate circuitvoltage control. In accordance with a preferred embodiment of theinvention, the current reference in the direction of the q axis isgenerated by using a direct voltage control value U_(dc,ref) of theintermediate circuit and the determined voltage U_(dc) of theintermediate circuit of the inverter. The current reference itself isgenerated by subtracting the measured voltage of the intermediatecircuit from the direct voltage reference value and by conducting theobtained difference variable to the intermediate circuit voltagecontroller, from the output of which the current reference i_(q,ref) isobtained, as can be seen from the block diagram of FIG. 2.

As described earlier, the measured currents have been converted to thesame coordinate system as the generated current references. Thus, thecomparison of the currents mentioned makes sense, and on the basis ofthe currents, the control circuits are further implemented in accordancewith the invention, the torque reference and the absolute valuereference of the flux linkage being obtained as the outputs of thecontrol circuits. The actual values i_(d) and i_(q) of the current ofthe synchronous coordination system must be low-pass-filtered before thecalculation of the difference variable. In FIG. 2, the low-passfiltering is implemented by means of filtering members 8 and 9.

Low-pass filtering the currents i_(d) and i_(q) provideslow-pass-filtered current vector components i_(d,lpf) and i_(q,lpf).Low-pass filtering removes the switching-frequency oscillation of thecurrent. An intense effect of the switching-frequency currentoscillation on the absolute value reference of the flux linkage and thetorque reference would prevent purposeful operation of the controlmethod. Conventional discreet filter algorithms can be used for thelow-pass filtering of currents.

In accordance with the invention, a torque reference t_(e,ref) isgenerated by comparing the current reference i_(q,ref) in the directionq with the current vector component i_(q,lpf) in the direction q.Preferably, the torque reference is generated in such a way that thedifference between the current reference in the direction q and thelow-pass filtered current vector component in the direction q isconducted to a current controller in the direction q, whereby the torquereference t_(e,ref) to be used is obtained from the output of thecurrent controller. In other words, the current controller 3 tends tominimize the difference between the current reference in the direction qand the low-pass filtered current vector component in the direction q byaffecting the output currents of the inverter through the torquereference.

In the method according to the invention, the absolute value reference|{overscore (ψ)}|_(ref) of the flux linkage is generated for the use ofthe DTC block by comparing the current reference i_(d,ref) in thedirection d with the low-pass-filtered current vector componenti_(d,lpf) in the direction d. Preferably, the absolute value referenceof the flux linkage is generated by subtracting the low-pass-filteredcurrent vector component i_(d,lpf) in the direction d from the currentreference i_(d,ref) in the direction d. The difference obtained from thesubtraction is further conducted to a current controller in thedirection d, whereby the absolute value reference |{overscore(ψ)}|_(ref) of the flux linkage to be used is obtained from the outputof the current controller. In other words, current vector components inthe direction d are used to control the size of the flux linkage, andthe absolute value reference of the flux linkage is further used as avariable on the basis of which the switching commands of the inverterare generated by using the DTC method known as such and the optimumswitching table related thereto. The above-described controllers 1, 2, 3and 4 are typically controllers operating by means of a PI algorithm,but it is to be noted that the controller algorithm to be used can beany algorithm implementing the control in a purposeful manner.

As mentioned earlier, the flux linkage absolute value and torque signalsgenerated by the outermost control loops are used in accordance with theinvention to form switching commands of power switches in accordancewith the DTC principle. The DTC block 5 contains a model of a motor or acorresponding load in a known manner. On the basis of this model, thevalues of the flux linkage and the torque are calculated from the load.

The DTC block 5 in FIG. 2 also comprises the power switches used in theinverter, whereby the voltages generated by the power switches formdirectly the output of the block. As shown in FIG. 2, currents i_(A),i_(C) of two phases are measured in accordance with the invention fromthe output of the block 5.

The DTC block further comprises innermost control loops that are thehysteresis controls of the flux and the torque. These controls areperformed by means of real values obtained from the motor model andreference variables generated by synchronous power controllers,according to which variables the inner controllers tend to adjust thevariables obtained from the model.

The optimum switching table is used in a known manner in such a way thaton the basis of the results of hysteresis comparisons used in saidhysteresis controls, the most purposeful switching command combinationas regards the control, at each particular moment of control, i.e. thevoltage vector command of the output, is selected. Subsequently, thiscommand is implemented by using separate control circuits controllingpower switches of the output.

In accordance with a preferred embodiment of the invention, thegenerated output voltage is low-pass-filtered. This low-pass filteringis shown in FIG. 2 as a filter 10. The filter 10 is typicallyphase-specific and can be of the type of an L filter formed of oneinductive component, an LC filter formed of an inductive component and acapacitive component, or for example an LCL filter formed of twoinductive components and one capacitive component. The use of a filterin connection with an inverter is extremely recommendable, because thecurve form of the current to be supplied to the network must be ratheraccurately similar to that of a sinusoid.

In the above, the method has been described in connection with aninverter, but it can also be applied to other alternate current devices,such as to electric motor use provided with an LC filter.

It will be obvious to a person skilled in the art that as the technologyadvances, the basic idea of the invention can be implemented in aplurality of ways. The invention and its embodiments are thus notlimited to the above examples but can vary within the claims.

1. A method in connection with an inverter, the inverter comprising adirect voltage intermediate circuit, an optimum switching table andoutput power switches, characterized by the method comprising the stepsof defining a synchronous rectangular dq coordinate system; measuringthe phase currents (i_(A), i_(C)) of the inverter outputs; convertingthe phase currents (i_(A), i_(C)) of the inverter to a synchronous dqcoordinate system in order to achieve synchronous rectangular currentvector components (i_(d), i_(q)); low-pass-filtering the synchronouscurrent vector components (i_(d), i_(q)) in order to achievelow-pass-filtered current vector components (i_(d,lpf), i_(q,lpf));generating a current reference (i_(q,ref)) in the direction of the qaxis of the dq coordinate system; generating a current reference(i_(d,ref)) in the direction of the d axis of the dq coordinate system;generating a torque (t_(e,ref)) reference by comparing the currentreference (i_(q,ref)) in the direction q with a low-pass-filteredcurrent vector component (i_(q,lpf)) in the direction q; generating anabsolute value reference (|{overscore (ψ)}|_(ref)) of the flux linkageby comparing the current reference (i_(d,ref)) in the direction d with alow-pass-filtered current vector component (i_(d,lpf)) in the directiond; and forming switching commands of the power switches on the basis ofthe torque reference (t_(e,ref)) and the flux reference (|{overscore(ψ)}|_(ref)) by using an optimum switching table (5).
 2. A methodaccording to claim 1, characterized by the generation of the currentreference (i_(q,ref)) in the direction of the q axis of the dqcoordinate system comprising the steps of forming the direct voltagereference value (U_(dc,ref)) of the intermediate circuit; determiningthe voltage (U_(dc)) of the intermediate circuit of the inverter; andcomparing the direct voltage reference value (U_(dc,ref)) with thevoltage (U_(dc)) of the intermediate circuit to provide the currentreference (i_(q,ref)) in the direction q of the synchronous coordinatesystem.
 3. A method according to claim 2, characterized by thegeneration of the current reference (i_(q,ref)) in the direction of theq axis comprising the steps of subtracting the voltage (U_(dc)) of theintermediate circuit from the direct voltage reference value(U_(dc,ref)) to generate a difference variable; and conducting thegenerated difference variable to an intermediate circuit controller (1),whereby the current reference (i_(q,ref)) in the direction q is obtainedfrom the output of the intermediate voltage controller.
 4. A methodaccording to claim 1, characterized by the generation of the currentreference (i_(d,ref)) in the direction of the d axis of the dqcoordinate system comprising the steps of generating a reactive powerreference value (q_(ref)); estimating the reactive power (q_(est)) ofthe inverter; and comparing the reactive power reference value with theestimated reactive power of the synchronous current coordinate system toprovide the current reference (i_(d,ref)).
 5. A method according toclaim 4, characterized by the generation of the current reference(i_(d,ref)) in the direction of the d axis comprising the steps ofsubtracting the estimated reactive power (q_(est)) of the inverter fromthe reactive power reference value (q_(ref)) to provide a differencevariable; and conducting the generated difference variable to thereactive power controller (2) to provide the current reference(i_(d,ref)) in the direction d.
 6. A method according to claim 1,characterized by the generation of the torque (t_(e,ref)) comprising thestep of generating the torque (t_(e,ref)) by subtracting thelow-pass-filtered current vector component (i_(q,lpf)) in the directionq from the current reference (i_(q,ref)) in the direction q.
 7. A methodaccording to claim 6, characterized by the generation of the torquereference (t_(e,ref)) comprising the step of conducting the differencebetween the torque reference (i_(q,ref)) in the direction q and thelow-pass-filtered current vector component (i_(q,lpf)) in the directionq to a current controller (3) in the direction q, whereby the torquereference (t_(e,ref)) is obtained from the output of the currentcontroller.
 8. A method according to claim 1, characterized by thegeneration of the absolute value reference (|{overscore (ψ)}|_(ref)) ofthe flux linkage comprising the step of generating the absolute valuereference (|{overscore (ψ)}|_(ref)) of the flux linkage by subtractingthe low-pass-filtered current vector component (i_(d,lpf)) in thedirection d from the current reference (i_(d,ref)) in the direction d.9. A method according to claim 8, characterized by the generation of theabsolute value reference (|{overscore (ψ)}|_(ref)) of the flux linkagecomprising the step of conducting the difference of the currentreference (i_(d,ref)) in the direction d and the low-pass-filteredcurrent vector component (i_(d,lpf)) in the direction d to a currentcontroller (4), whereby the absolute value reference (|{overscore(ψ)}|_(ref)) of the flux linkage is obtained from the output of thecurrent controller (4).
 10. A method according to any one of thepreceding claims, characterized by the switching commands of the powerswitches being implemented to generate an output voltage and thegenerated output voltage being low-pass-filtered.
 11. A method accordingto claim 1, characterized by the conversion of the phase currents of theinverter to the synchronous dq coordinate system comprising the steps ofdetermining from the measured phase currents rectangular components(i_(x), i_(y)) of the current vector of the inverter in the stationaryxy coordinate system; and converting the rectangular components (i_(x),i_(y)) of the current vector of the stationary coordinate system to thesynchronous dq coordinate system to provide rectangular current vectorcomponents (i_(d), i_(q)).